Abstract-The way-below relation defined on dcpos is a important concept, it is the basis for the continuity of dcpos. And a poset is also a category, so we want to generalize the concept of way-below to category and further define continous category. In this paper, we give the way-below relation on arbitrary small categories and even a topos and introduce the concept of continuous category and obtain some correspontive characterizations by mean of the diagram proof. This also shows diagram method can be used to reconstruct the classical order theory in an arbitrary elementary topos.
In this paper, based on the definition of group object, the definition of action of group object on arbitrary object in a topos is given, some equivalent characterizations are also obtained.
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